标题： 密度代数上的几何构造 显示英文标题

标题： Geometric constructions on the algebra of densities

摘要： 密度代数$\Den(M)$是与给定流形或超流形$M$相关联的交换代数。 我们之前在研究Batalin--Vilkovisky几何时介绍了这个代数。 代数$\Den(M)$按实数分级，并具有自然的不变标量积。 这导致了重要的几何后果，并应用于原始流形上的几何构造。 特别地，存在代数$\Den(M)$的导子的分类定理。 它允许在(超)流形$M$上各种权重的向量密度上自然地定义括号运算，类似于经典Frölicher--Nijenhuis定理关于微分形式代数的导子如何导致Nijenhuis括号。 可以将这种分类从$\Den(M)$上的“向量场”(导子)扩展到“多向量场”。 这导致了一个显著的结果，即$M$上任意偶数泊松结构在密度代数上具有规范提升。 (后两个陈述是由我们的学生A.Biggs获得的。) 这与之前研究的奇数泊松结构的情况形成鲜明对比，在奇数泊松结构中，需要额外的数据来进行这种提升。 显示英文摘要

摘要： The algebra of densities $\Den(M)$ is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra $\Den(M)$ is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra $\Den(M)$. It allows a natural definition of bracket operations on vector densities of various weights on a (super)manifold $M$, similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from "vector fields" (derivations) on $\Den(M)$ to "multivector fields". This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.